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Differentiation Rules

Differentiation Rules Problem Solving


Consider a function f(x)f(x) satisfying f(x)=f(4x2) f(x)=f(4x-2) for all real values xx. If f(x)f(x) is differentiable for all xx and f(4)=40,f'(4)=40, what is the value of f(54)?f'(54)?

Given ddxlnx+2x7=f(x)2(x+2)(x7),\frac{d}{dx} \ln \frac{x+2}{\sqrt{x-7}}=\frac{f(x)}{2(x+2)(x-7)}, what is f(x)?f(x)?

The polynomial P(x) P(x) satisfies the following identity: P(P(x)+x)=11(P(x)+x)24(P(x)+x)+5. P \left( P(x) + x \right) = 11 \left( P(x) + x \right)^2 - 4 \left( P(x) + x \right) + 5 . What is the value of P(6)? P'(6)?

Given g(x)=xsin1(x24)+576x2,g(x)=x \sin^{-1} \left(\frac{x}{24}\right)+\sqrt{576-x^2}, what is the value of g(12)g'(12)?

Let g(x)g(x) be the inverse function of a differentiable function f(x).f(x). If f(3)=7f(3)=7 and f(3)=12,f'(3)=\frac{1}{2}, what is the value of limh0g(7+h)g(7)h?\lim_{h \to 0} \frac{g(7+h)-g(7)}{h}?


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